Renormalization in a classical lattice field theory

Abstract

We formulate and describe a renormalization-group transformation for classical \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$-field theory on a lattice. The main idea is to divide the angle variables of the oscillators into the fast ones (large momenta) and the slow ones (low momenta) and to average over the fast ones. This results in an effective Hamiltonian for the remaining slow modes, which can be compared with the starting Hamiltonian. We derive fixed-point conditions and obtain a scaling law for those classical solutions for which the renormalization step can be iterated. There is a striking resemblence between our classical treatment and the analogous procedure in quantum field theory, which we discuss in some detail.